Properties

Label 2-224-224.13-c2-0-5
Degree $2$
Conductor $224$
Sign $0.175 - 0.984i$
Analytic cond. $6.10355$
Root an. cond. $2.47053$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1.91 − 0.577i)2-s + (−2.09 − 0.869i)3-s + (3.33 + 2.21i)4-s + (−0.289 − 0.697i)5-s + (3.51 + 2.87i)6-s + (4.51 + 5.35i)7-s + (−5.10 − 6.16i)8-s + (−2.71 − 2.71i)9-s + (0.150 + 1.50i)10-s + (−4.20 − 10.1i)11-s + (−5.07 − 7.54i)12-s + (−5.80 + 14.0i)13-s + (−5.54 − 12.8i)14-s + 1.71i·15-s + (6.20 + 14.7i)16-s − 6.67·17-s + ⋯
L(s)  = 1  + (−0.957 − 0.288i)2-s + (−0.699 − 0.289i)3-s + (0.833 + 0.553i)4-s + (−0.0578 − 0.139i)5-s + (0.586 + 0.479i)6-s + (0.644 + 0.764i)7-s + (−0.637 − 0.770i)8-s + (−0.301 − 0.301i)9-s + (0.0150 + 0.150i)10-s + (−0.382 − 0.922i)11-s + (−0.422 − 0.628i)12-s + (−0.446 + 1.07i)13-s + (−0.396 − 0.918i)14-s + 0.114i·15-s + (0.387 + 0.921i)16-s − 0.392·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 224 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.175 - 0.984i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 224 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.175 - 0.984i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(224\)    =    \(2^{5} \cdot 7\)
Sign: $0.175 - 0.984i$
Analytic conductor: \(6.10355\)
Root analytic conductor: \(2.47053\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{224} (13, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 224,\ (\ :1),\ 0.175 - 0.984i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.360039 + 0.301650i\)
\(L(\frac12)\) \(\approx\) \(0.360039 + 0.301650i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.91 + 0.577i)T \)
7 \( 1 + (-4.51 - 5.35i)T \)
good3 \( 1 + (2.09 + 0.869i)T + (6.36 + 6.36i)T^{2} \)
5 \( 1 + (0.289 + 0.697i)T + (-17.6 + 17.6i)T^{2} \)
11 \( 1 + (4.20 + 10.1i)T + (-85.5 + 85.5i)T^{2} \)
13 \( 1 + (5.80 - 14.0i)T + (-119. - 119. i)T^{2} \)
17 \( 1 + 6.67T + 289T^{2} \)
19 \( 1 + (4.52 - 10.9i)T + (-255. - 255. i)T^{2} \)
23 \( 1 + (-23.1 - 23.1i)T + 529iT^{2} \)
29 \( 1 + (12.5 - 30.2i)T + (-594. - 594. i)T^{2} \)
31 \( 1 - 43.5iT - 961T^{2} \)
37 \( 1 + (37.6 - 15.5i)T + (968. - 968. i)T^{2} \)
41 \( 1 + (33.6 - 33.6i)T - 1.68e3iT^{2} \)
43 \( 1 + (-2.11 - 5.11i)T + (-1.30e3 + 1.30e3i)T^{2} \)
47 \( 1 - 80.2T + 2.20e3T^{2} \)
53 \( 1 + (2.85 + 6.88i)T + (-1.98e3 + 1.98e3i)T^{2} \)
59 \( 1 + (-8.20 - 19.8i)T + (-2.46e3 + 2.46e3i)T^{2} \)
61 \( 1 + (82.3 + 34.0i)T + (2.63e3 + 2.63e3i)T^{2} \)
67 \( 1 + (-17.9 + 43.3i)T + (-3.17e3 - 3.17e3i)T^{2} \)
71 \( 1 + (87.4 - 87.4i)T - 5.04e3iT^{2} \)
73 \( 1 + (-12.3 + 12.3i)T - 5.32e3iT^{2} \)
79 \( 1 + 68.3iT - 6.24e3T^{2} \)
83 \( 1 + (-10.3 + 24.9i)T + (-4.87e3 - 4.87e3i)T^{2} \)
89 \( 1 + (82.5 + 82.5i)T + 7.92e3iT^{2} \)
97 \( 1 - 94.9iT - 9.40e3T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−11.95204142174723924611123981673, −11.30391413301975418960565488313, −10.52803456589551675180884038825, −8.977095133568090567377937022078, −8.669420837364674598300495208478, −7.25970063753230113155816992200, −6.26986438407502091316229187674, −5.11156092603371937168363048624, −3.10469252545269730475044311523, −1.46628023341372386160820990781, 0.38749757554112914560842224027, 2.41851533672007137311144122825, 4.67001151192167377579447554447, 5.59677431376564870725506811977, 7.01879225436985602679376916865, 7.71182650782007005154984296245, 8.826357761927328102315275261519, 10.14651079746007594979349012857, 10.70012438868628480407372109331, 11.34426966405322056457157391825

Graph of the $Z$-function along the critical line